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Radiation physics constraints on global 
warming: C02 increase has little effect 



By Denis G. Rancourt 
Former physics professor, University of Ottawa, Ottawa, Canada. 



Posted on June 3, 201 1 : 

http://climateguy.blogspot.com/201 1/06/radiation-physics-constraints-on-global.html 

(Small clarifications added and several typos corrected here in this December 3, 2011, pdf version.) 



Abstract — I describe the basic physics of planetary radiation balance and 
surface temperature, using the simplest model possible that is sufficiently realistic 
for correct evaluations of predicted surface temperature response sensitivities to 
the key Earth parameters. The model is constrained by satellite absolute integrated 
intensity and spectroscopic measurements and the known longwave absorption 
cross section of C02 gas. I show the predicted Earth temperature for zero 
atmospheric resonant absorption of longwave radiation (no greenhouse effect in 
the otherwise identical atmosphere) to be -46°C, not -19°C as often wrongly 
stated. Also, the net warming effect from the atmosphere, including all 
atmospheric processes (not just greenhouse forcing), without changing anything 
else (except to add the removed atmosphere) is +18°C, not the incorrect textbook 
value of +33°C. The double-layer atmosphere model with no free parameters 
provides: (a) a mean Earth surface temperature of +17°C, (b) a post-industrial 
warming due only to C02 increase of 5T = 0.4°C, (c) a temperature increase from 
doubling the present C02 concentration alone (to 780 ppmv C02; without water 
vapour feedback) equal to 8T = 1 .4°C, and (d) surface temperature response 
sensitivities that are approximately two orders of magnitude greater for solar 
irradiance and for planetary shortwave albedo and longwave emissivity than for 
the atmospheric greenhouse effect from C02. All the model predictions robustly 
follow from the straightforward underlying assumptions without any need for 
elaborate global circulation models. The same longwave optical saturation that 
provides such a large radiative warming of the planet surface also ensures that the 
warming effect from increasing C02 concentration is minimal. I conclude with 
suggested implications regarding warming alarmism, errors by sceptics, research 
funding, and scientific ignorance regarding climate feedbacks. 



Rancourt on radiation physics - C02 little effect 



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Introduction 



Historically, the greatest ability of the physicist has been to perform simple calculations 
that capture the essential features of a physical phenomenon in order to correctly 
elucidate the underlying principal causes. This is the ultimate "What is going on?" 
challenge. 

Too many practicing physicists, in the many areas where physics is applied, have lost or 
never had this ability. Instead, they have been incorporated into the enterprise of using 
computers to simulate reality using questionable selections of approximate or invalid 
algorithms in large simulation programs. 

These programs develop lives of their own, as careers and reputations are invested in 
their incremental development and in their predictions. A pathological optimism envelops 
the practitioners with the illusion that their algorithms capture complex features in some 
"average" or "effective" way and efforts are made to demonstrate this in so-called 
"validation" exercises rather than perform simple calculations that would demonstrate the 
algorithms themselves to be wrong for the intended application. 

Physicists have largely abandoned their gadfly role of fundamentally challenging broad 
interpretive schemes in order to serve and benefit from career-enhancing collective 
enterprises, often dressed in elaborate conceptual edifices and often supported by 
computer simulations. 

I believe this situation is playing itself out today in climate modelling science. As a 
physicist, if on close examination I can't understand what the C02 warming alarmism is 
about and I can't get any of my colleagues to explain it - without computer-black-box 
magic, in published papers or elsewhere - then I am not going to believe it. 

At its core, planetary surface temperature is a macroscopic radiation balance phenomenon 
that has been understood for one hundred years or so. If global warming alarmism is 
justified then it must be possible to explain why it is justified in simple terms and without 
appealing to faith or authority for any essential point in the argument. 

I've tried to do this, as honestly and openly as possible, and I have asked my peers to find 
any errors. I believe the present article to be error-free and to conclusively show that we 
should not be focussed on C02 if we are concerned about the planet's surface 
temperature. I am additionally of the opinion that we should not be concerned about the 
planet's surface temperature. 

Regarding the sceptic-warmist debate, my conclusion is: The sceptics say many incorrect 
things but they are right whereas the warmists say many correct things but they are 
wrong. The skeptics appear to be motivated by skepticism whereas the warmists appear 
to be motivated by conformism. The skeptics' incorrect things have been used to discredit 
the skeptics and the warmists' correct things have been used to mask a lie. 



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Simplest models with essential features 



My goal is to construct the simplest possible models of planetary radiation balance, 
realistic enough to capture the essential global average features using different physical 
assumptions about the type of atmosphere. 

I take the planet to be a perfect sphere with a smooth and homogeneous surface and to 
have a thin (compared to the planet radius, but longwave optically thick) atmosphere. The 
planet is uniformly irradiated by a distant sun. 

The incident intensity (in Watts per square -meter, W/m ) of "shortwave" radiation 
(largely visible light) from the sun at the planet is the so-called solar constant, I s , where 
for Earth I s = 1366 W/m 2 (having a real seasonal variation in magnitude from 1412 to 
1321 W/m , or 6.7% of its average value). 

Different parts of the planet's surface receive different intensities of incident shortwave 
radiation. This is because the surfaces at different latitudes receive the incident rays at 
different angles and because half of the planet's surface is shielded from all incident rays 
(only one hemisphere is exposed to the sun at any given time). 

Rather than deal with the latter complexity of non-uniform irradiation, instead, as is 
commonly done, we take the entire planet's surface to be uniformly irradiated with an 
intensity equal to the corresponding average solar constant. The correct average solar 
constant is <I S > = (1/4)I S = 341.5 W/m 2 , as is well known and easy to calculate. 

In my models, therefore, every part of the planet's surface is identical in terms of the 
radiation balance conditions. Each part of the planet's surface represents what is 
happening on average, in terms of radiation balance, and of the planet properties which 
we take to be the Earth's average properties. 



Basic concepts and Earth with no atmosphere 

Of all the incident shortwave solar radiation that strikes the planet a fraction is reflected 
back into outer space without being absorbed by any part of the planet (surface or 
atmosphere). This fraction (from zero to one) of the incident shortwave solar radiation 
energy that is reflected out from the planet is called the planet's (Bond) albedo. 

The reflected outgoing shortwave radiation need not have the same spectral distribution 
(radiation intensity versus radiation frequency or wavelength) as the incoming incident 
solar shortwave radiation because the amount of absorption/reflection can be (and 
generally is) dependent on wavelength. The albedo is the net energy fraction that is 
reflected. 



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Modern satellite spectroscopic measurements can quantify the solar constant and the 
amount of out-reflected shortwave radiation, can resolve these radiations from longwave 
thermal radiation, and can measure continuously in orbit to obtain planet- wide averages. 

Satellite measurements allow us to conclude that the average Earth albedo is <a> = 0.30 
[1]. Arguably-more-direct and reliable Earth-based so-called "Earthshine observation" 
measurements give <a> = 0.297(5) where, using scientific error notation, the latter means 
0.297 ± 0.005 [2]. There are daily changes in Earth's albedo (from large scale weather 
changes) of -5% and seasonal variations of -15% (from snow and ice cover, vegetation, 
and weather and cloud cover) [2]. 

The main source of heat on the planet is the planet's surface that absorbs shortwave solar 
radiation. The physical absorption process transforms the electromagnetic energy of the 
incident solar radiation into heat energy (vibrational energy of the surface's molecules). 
In the case with an atmosphere, the atmosphere also directly absorbs a fraction of the 
incident shortwave solar radiation. 

Any opaque body at any temperature above 0 K (i.e., having vibrating rather than 
motionless molecules) in turn emits electromagnetic radiation. The latter so-called 
"thermal" or "black-body" radiation has characteristics that depend on the body's 
"emitting surface" temperature. The spectral distribution of such emitted thermal 
radiation follows Planck's Law (modified to allow a wavelength-dependent emissivity). 
For the temperatures of interest the surface thermal radiation is longwave (or infra-red) 
radiation. 

The intensity I e (in W/m ) of the emitted thermal (here longwave) electromagnetic 
radiation coming from the effective emitting surface of a somewhat opaque body is given 
by the Stefan-Boltzmann law: 

I e = s a T 4 (eq.l) 

where T is the temperature of the emitting surface in K, o is the Stefan-Boltzmann 

8 2 4 

constant a = 5.6704 x 10" W/m K , and s is the "emissivity" of the emitting surface valid 
for the relevant emitted frequencies. 

The emissivity has a dimensionless value between zero and one. It is the fractional 
energy emission from the surface compared to the surface's emission if it were an ideal 
black body emitter, s = 1 for an ideal black body surface and s = 0 for an ideally 
reflective surface (i.e., a surface having an albedo of exactly 1). 

The global average emissivity (for the relevant longwave radiation), <s>, of Earth's 
surface is difficult to evaluate. It can be reasonably estimated by considering the known 
measured longwave emissivities of typical Earth surface materials, such as liquid water, 
vegetation, and sand. 

Let us next describe how the planet's mean surface temperature is established. 



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If the net radiant energy into the planet is larger than the net radiant energy escaping from 
the planet then the net received energy will heat the planet and increase its temperature. 
Likewise, if the net radiant energy out from the planet is larger than the net energy into 
the planet then the net loss of energy of the planet will cause the planet to loose heat and 
decrease its temperature. 

Therefore, in a "steady state" situation, after a certain planetary response time following 
any change affecting radiation balance, the temperature of the planet's longwave 
radiation emitting surface will stabilize at some value corresponding to the rate of 
energy-in being equal to the rate of energy-out and there will be planetary "radiation 
balance" at a stable planetary surface temperature. 

The net energy-in is the incident solar radiation minus the albedo loss. With no 
atmosphere, the net energy-out is the longwave emission energy from the planet's surface 
escaping into outer space. By setting in = out we can solve for the resulting radiation- 
balancing planet surface temperature. 

The corresponding radiation balance equation, therefore, can be written as: 
<I S > (1 - <a>) = <e> a T 4 (eq.2) 

Solving for the planet surface temperature T 0 (in K) for no atmosphere, eq.2 gives: 
T 0 = [ (1 - <a>)<I s > / <e>o ] 1/4 . (eq.3) 

At this point, virtually all researchers and authors have used <s> = 1 , usually without 
providing a stated justification. That is, they have assumed that the Earth's surface is an 
ideal black body emitter for longwave radiation. 

Using the latter assumption for <s> and (for now, wrongly) assuming that the Earth's 
mean albedo <a> is the same with and without its atmosphere (<a> = 0.30) eq.3 gives T 0 
= 254.8 K or minus (-) 18.3°C. Compared to the accepted actual mean global surface 
temperature of 14.0°C this would imply a total global atmosphere (greenhouse) effect 
warming on Earth of 32.3°C - corresponding to the repeatedly stated textbook nominal 
value of 33°C of greenhouse effect warming [3]. 

This surface temperature (nominally -19°C) is also the surface temperature (with <s> = 
1) of the bare planet (no atmosphere but preserving the same albedo) that would give the 
same total emission of longwave radiation presently escaping from Earth into outer 
space. 

Many authors have stated that this thus calculated nominal -19°C temperature is "the 
Earth's temperature as seen from outer space". The latter statement is incorrect because 
although the actual present integrated emission intensity would, via eq. 1 , give this 
temperature, the actual longwave emission spectrum of Earth is not a black-body 



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emission spectrum (i.e., does not follow Planck's Law, due to significant atmospheric 
absorption) and only a black-body-radiation spectrum can be interpreted as corresponding 
to an emitter's "temperature". 

Wrong textbook views of Earth warming 

The above described repeatedly stated textbook [3] value of 33°C of Earth greenhouse 
effect warming is wrong because this is not the correct predicted value of planet surface 
warming (or cooling) that occurs on turning on (or off) the greenhouse effect in an 
otherwise unchanged Earth atmosphere and otherwise unchanged Earth. 

I also taught the incorrect 33°C value in my university physics courses and repeated it in 
my 2007 critique of global warming [4]. Wikipedia is no exceptions [5]. American 
Geophysical Union (AGU) press releases typically announce [6]: 

"Overall, the greenhouse effect warms the planet by about 33 °C, turning it from 
a frigid ice-covered ball with a global average temperature of about -1 7 °C, to 
the climate we have today. Heat-absorbing components contribute directly to that 
warmth by intercepting and absorbing energy passing through the atmosphere as 
electromagnetic waves. " 

In describing the "physical science basis" the Intergovernmental Panel on Climate 
Change (IPCC) in its 2007 "Contribution of Working Group I to the Fourth Assessment 
Report" (incorrectly, see below) put it this way [7]: 

"The energy that is not reflected back to space is absorbed by the Earth 's surface 
and atmosphere. This amount is approximately 240 Watts per square metre (W 
m—2). To balance the incoming energy, the Earth itself must radiate, on average, 
the same amount of energy back to space. The Earth does this by emitting 
outgoing longwave radiation. Everything on Earth emits longwave radiation 
continuously. That is the heat energy one feels radiating out from a fire; the 
warmer an object, the more heat energy it radiates. To emit 240 W m—2, a surface 
would have to have a temperature of around -19°C. This is much colder than the 
conditions that actually exist at the Earth 's surface (the global mean surface 
temperature is about 14°C). Instead, the necessary -19°C is found at an altitude 
about 5 km above the surface. 

The reason the Earth 's surface is this warm is the presence of greenhouse gases, 
which act as a partial blanket for the longwave radiation coming from the 
surface. This blanketing is known as the natural greenhouse effect. " 

The scientists at RealClimate.org also (incorrectly, see below) use this 33°C number in 
their interpretations [8]: 



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"Since we are looking at the whole of the present-day greenhouse effect (around 
33 C), it is not surprising that the radiative forcings are very large compared to 
those calculated for the changes in the forcing. The factor of~2 greater 
importance for water vapour compared to C02 is consistent with the first 
calculation. " 

Virtually all mainstream science and teaching has accepted and parrots this idea that the 
planetary greenhouse effect on Earth causes a warming of approximately 33°C. 

In all of these sources the assumption <s> = 1 is virtually never explicitly justified. It is 
important to provide a justification because, at first glance, the assumption appears to 
violate Kirchoff s Law of radiation physics. 

Kirchoff s Law of radiation physics says generally that the larger the reflectivity the 
smaller the emissivity. More precisely, Kirchoff s Law is expressed for a given 
wavelength X as: 

1 - a(X) = s(k). (eq.4) 

It is essential to note that the law holds at each wavelength (and direction) of radiation 
but that albedo at one wavelength need not be related to emissivity at a different 
wavelength. 

On Earth, the relevant mean (Bond) albedo is for shortwave radiation (solar radiation, 
largely visible) and has a value <a> = 0.30 whereas the needed emissivity is for 
longwave radiation (infra-red or thermal Earth-emission radiation) such that <s> can 
have a value significantly different from the value 0.70 incorrectly predicted by eq.4. 

We must therefore appeal to measurements of s for representative Earth surface 
materials. A main Earth surface material is water. The longwave emissivity of water is 
almost 1 . This is understandable because water is almost perfectly absorbing in the 
infrared. Dry rocks and sand also have near-one values of their longwave emissivities, 
that is values of -0.91-0. 92. Any vegetation coverage of dry soil significantly increases 
the value of the emissivity, given the water content of vegetation. For example, green 
grass has emissivity in the range -0.97-0. 99 [9]. 

This is why it is not unreasonable to use <s> ~ 1 for our ocean, lake and vegetation- 
covered Earth. Therefore, the assumed value of unity for emissivity is not significantly in 
error for our purposes. 

Similarly, note that the above calculation leading to the nominal -19°C surface 
temperature is not for an Earth without its atmosphere but that is otherwise unchanged; 
because such an instantaneously bared Earth would not retain its albedo of 0.30. Indeed, a 
large contribution to Earth's albedo is from clouds and the atmosphere itself - it is not a 
purely planet-surface albedo. 



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With no atmosphere we should use the albedo of the Earth's present solid surface, in its 
present state. The latter shortwave albedo <a s > has been measured by satellite and is 
23/(23+161) = 0.125 ([1]: Fig.l). This gives (eq.3) the significantly higher no-atmosphere 
mean surface temperature of T 0 = 269.4 K (or -3.7°C), for a total atmosphere warming 
effect without changing anything else on the present Earth of +18°C, not +33°C. The 
correct predicted surface temperature of an Earth with no atmosphere but otherwise 
unchanged is -4°C. 

Of course without an atmosphere there would be no vegetation, etc., and significantly 
more snow and ice cover, thereby increasing the surface albedo. The latter changes are 
difficult to predict and obviously have not been measured by satellite. In any case, the 
relevant question for the present discussion is "What is the net warming effect from the 
atmosphere, including all its processes, without changing anything else?" The answer is 
+18°C,not+33°C. 

It should be clear therefore that the oft-repeated nominal -19°C surface temperature both 
is not the Earth temperature without an atmosphere (but otherwise unchanged) and is not 
"the temperature of Earth as seen from outer space". It is a nonsense number arising from 
an incorrect application of eq. 1 . 

But error in most warming establishment spin is even larger than that because the 
presence or absence of the atmosphere is not equivalent (in terms of global radiation 
balance) to the presence or absence of an atmospheric greenhouse forcing, by any means. 
In other words, in terms of the planetary radiation balance, removing the atmosphere and 
removing the greenhouse action of the greenhouse gases in the atmosphere have 
dramatically different effects. 

This is shown below and is because the atmosphere impacts surface temperature by much 
more than only via greenhouse forcing. And the other impacts cause surface cooling 
rather than warming. These other impacts are: (1) direct absorption by the atmosphere of 
incident shortwave solar radiation (78 W/m 2 ; [1]), (2) increased albedo from clouds and 
atmosphere (0.30 vs. 0.125), (3) surface cooling via atmospheric thermals (17 W/m 2 ; [1]), 
and (4) surface cooling via the evaporation/condensation via the water cycle (80 W/m 2 ; 

[1]). 

I show below that, as a result, the correct Earth surface temperature in the absence of 
greenhouse forcing but with an otherwise unchanged atmosphere is a freezing -46°C. 
This means that "global warming" from atmospheric resonant scattering of infrared 
radiation on Earth is +60°C, from -46°C to +14°C, not +33°C. 

Note that despite the large (~60°C) predicted atmospheric greenhouse warming on Earth 
this includes the total planet greenhouse effect whereas C02 absorption is presently 
saturated (see below), such that a large C02 change impact is not implied. It is essential 
to recognize that a large overall planetary greenhouse warming does not imply a 
significant warming from increasing the concentration of C02. Indeed only a much 
smaller overall planetary greenhouse warming could give rise to a large warming effect 



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from increasing the C02 concentration in the atmosphere. The two factors (total warming 
versus sensitivity to C02 increase) are anti-correlated via the phenomenon of optical 
saturation (see below). 

Earth with homogeneous atmosphere 

Consider first the simplest atmosphere; an atmosphere which is uniform in its 
temperature, composition and density and that attains its own steady state temperature by 
balancing its own net in and out radiative and other fluxes of energy. Let the temperature 
of the atmosphere (in K) be denoted Tp and the temperature of the planet surface be 
denoted T a . 

In this case, the longwave emission of the atmosphere (eq.l) up and out is equal to its 
longwave emission down and in (which is fully absorbed by the planet surface, <s> ~ 1). 
In addition, since the atmosphere layer emits both up and down, it has a thermal emission 
surface of 2<S in area for every surface of area & of the planet surface. 

In our homogeneous atmosphere model we include convective and air-mass thermals 
heating and water latent heat heating of the atmosphere, and evaporation and thermals 
cooling of the planet surface. 

Consequently, the following energy flux balance equation must hold for the planet's 
surface: 

A + p-a-C-D = 0... (eq.5) 
where 

• "A" is the average intensity of incident shortwave solar radiation directly 
absorbed by the Earth's surface, 161 W/m 2 [1] 

• (3 = s at a Tp 4 is the average longwave emission intensity from the atmosphere to 
the Earth and e at is the longwave emissivity of the atmosphere 

• a = <s> a T„ 4 is the average longwave emission intensity from the Earth surface 
and <s> is the longwave emissivity of Earth's surface, as defined above 

• "C" is the (mean global) upward energy flux intensity from thermals, taking heat 
from the surface and delivering it into the atmosphere, 17 W/m [1] 

• "D" is the (mean global) upward energy flux intensity from the water cycle 
(evaporation and latent heat condensation/freezing), taking heat from the surface 
and delivering it into the atmosphere, 80 W/m 2 [1] 



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Eq.5 represents all the radiation and other energies absorbed by and emitted by the planet 
surface. If the atmospheric temperature (of the assumed homogeneous atmosphere) was 
known, eq.5 would lead to a predicted surface temperature. 

Likewise, the following energy flux balance equation must hold for the planet's 
atmosphere: 

B + (1 - a at )a- 2(3 + C + D = 0 ... (eq.6) 
where 

• "B" is the intensity of incident shortwave solar radiation directly absorbed by the 
atmosphere, 78 W/m 2 [1] 

• a at is the longwave "albedo" of the atmosphere, the fraction (from zero to one) of 
longwave radiation intensity incident on the atmosphere that is not absorbed by 
the atmosphere, referred to elsewhere as the atmosphere's transmission coefficient 
<t e > [10] 

The global average value of a at is known from satellite measurements to be a at = 40 W/m 
/ 396 W/m 2 = 0.10 [1]. In addition, eq.4 implies s at = 1 — a at = 0.90 (both are longwave 
values for the atmosphere). 

This system of two equations (eqs.5 and 6) is immediately solved for the two unknowns 
(a and (3) such as: 

a = (1 + a at )-'[2A + B - C - D] ... (eq.7) 
And gives: 

T a = 264 K (or -9°C) and ... (eq.8a) 
T p = 254 K (or -19°C). ... (eq.8b) 

It is also possible to "turn off all greenhouse effect forcing by setting a at = 1 (i.e., letting 
a at approach a value of one). This gives a zero-greenhouse-warming Earth surface 
temperature of 

T a = 227 K (or -46°C) ... in the absence of greenhouse effect warming ... (eq.9) 

The latter value for the surface temperature of a non-greenhouse Earth is maintained for 
all multi-layer atmosphere models, with any number of atmosphere layers and with any 
distributions of energy deliveries to the different layers (energies B, C, and D partitioned 
to the different atmosphere layers). This is shown below, for all models using equally 
opaque (sufficiently optically thick) layers sharing the same value of a at . 



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Interpreting the homogeneous atmosphere prediction 



The above predicted Earth surface temperature T a = 264 K (or -9°C) is low compared to 
the accepted measured average of 14°C [11] and the predicted difference with the 
homogeneous atmosphere temperature Tp = 254 K (or -19°C) is only 10°C. 

This occurs because a single-layer homogeneous atmosphere does not allow the air closer 
to the surface to be warmer than air of higher altitude. Energy transport from the surface 
via thermals and the water cycle is considered to occur towards the full atmosphere rather 
than be concentrated near the surface. 

We can consider that the near-surface air is part of an effective surface and that the 
thermal and water cycle energy deliveries occur near the surface (e.g., fog, wind-mixing 
of thermal transport, etc.) and are cyclically confined to a near-surface region by simply 
setting C = D = 0. This implies that no thermal and water cycle energy exchanges occur 
with the atmosphere -proper that is taken to be distinct from a near-surface atmosphere 
layer considered to be part of the effective surface. 

When, in this way, only radiative heating and cooling are considered we obtain: T a = 283 
K (or +10°C) and Tp = 239 K (or -34°C). These values are close to actual values for 
Earth. This suggests that multi-layer atmosphere models are needed for sufficient realism. 

Earth with double-layer atmosphere 

Let the planet surface be denoted a, the inner atmosphere layer be denoted [3 and the outer 
atmosphere layer be denoted y. For simplicity, let the direct incident solar, thermals, and 
water cycle energy inputs to the atmosphere be divided between and delivered equally to 
the two atmosphere layers. 

Therefore, the energy flux balance condition for a is: 

A - C - D - a + p + a at y = 0 ... (eq.10) 

where 

• Y = s at cj T y 4 is the longwave emission intensity from the y-layer of the 
atmosphere to the Earth and T 7 is the uniform temperature of the y-layer 

Each atmosphere layer emits equal longwave emission intensities both up (towards 
space) and down (towards Earth). Each atmosphere layer is equally optically opaque, 
with the same values of a at . The latter arises because of the high degree of longwave 
optical opaqueness (high degree of resonant absorption over saturation, ~4 orders of 
magnitude or so) of the total atmosphere. 

And, the energy flux balance equations for [3 and y are, respectively: 



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B/2 + C/2 + D/2 + (1 - a at )a -2p + (1 - a at )y = 0 ... (eq. 1 1) 
and 

B/2 + C/2 + D/2 + a at (l - a at )a + (1 - a at )p - 2y = 0 ... (eq. 12) 
These equations (eqs.9, 10, 11) give: 

a = (l/2)(3 + 2a at - a at 2 )(l + a at )" 2 [2A + B - C - D] ... (eq.13) 
and 

T a = 290 K (or +17°C), ... (eq.l4a) 
T p = 280 K (or +7°C), and ... (eq.l4b) 
T Y = 251 K(or-22°C). ... (eq.l4c) 

These temperature values are close to actual values for Earth despite the remarkable 
simplicity of the model with no free parameters. This suggests that a radiation balance 
approach is correct despite ancillary complications related surface roughness, lapse rate 
constraints, Earth's rotation, non-uniform irradiation, thermal response times, an 
inhomogeneous atmosphere, and an inhomogeneous surface. It also suggests that the 
model is sufficiently realistic to calculate response sensitivities to changing its key 
parameters. 

We note that the only difference between eq.13 and eq.7 is in the pre factor to the 
intensity terms and that eq.13 gives the same no-greenhouse-effect (a at =1) prediction as 
eq.7: 

a = [A + B/2 - C/2 - D/2] ... (eq.15) 

giving T a (a at = 1) = 227 K (or -46°C) (same as eq.8). Indeed, the structure of the 
equations show that this is a robust result that would be the same for any number of 
optically opaque layers and using any division or distribution of direct incident solar, 
thermals, and water cycle energy inputs to the atmosphere layers. 

Eq.15 shows that under no-greenhouse-effect conditions half of B and of C and of D are 
longwave re-radiated back to Earth by the emitting atmosphere, irrespective of its 
vertically inhomogeneous structure. The other halves of B, C, and D are radiated out to 
space. 

Temperature change scenarios and sensitivity predictions 

In this section we use our double-layer atmosphere model to predict surface temperature 
responses to various C02 scenarios and to other changes in key physical parameters. 



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First we consider the model (eq.13) predictions for doubling the present atmospheric 
C02 concentration and for the post-industrial increase in atmospheric C02 concentration, 
without changing anything else. That is, in the absence of any water vapour feedback or 
any other such positive or negative feedback. 

For these calculations we must develop an equation that relates changes in atmospheric 
C02 concentration to corresponding changes in atmosphere albedo a at , equavalent to net 
longwave forward transmission through the atmosphere (denoted <t e > in a previous paper 
[10]). A given atmospheric C02 concentration (in ppmv, parts per million per volume) is 
denoted C co2 . 

The present known value of a at (-0.10) is significantly smaller than 1 and C02 longwave 
absorption predominantly occurs in a limited wavelength range (from -600 to -800 
wavenumber, 1/cm) centered on -15 jam (micro-meter wavelength), such that absorption 
saturation occurs in this main relevant C02 absorption band [12]. 

This implies that the induced change in a at is not simply (anti-)proportional to the 
considered change in C02 concentration (change in C C0 2) but instead is highly attenuated. 
Indeed, the decrease in a a t from an increase in C C0 2 arises not from an increased 
absorption at resonance but instead from increased absorption on the outer edges of the 
absorption band, thereby increasing the wavenumber- width of the absorption region 
corresponding to saturation absorption conditions (e.g., [12]: Fig. 2). 

Here, we derive the needed relation between a at and C C0 2 as follows. 

We take the main relevant C02 longwave absorption band to be mathematically 
represented by a Gaussian function having a height and width equal to the height and 
width of the actual (non-saturated) absorption cross section for the C02 band centered at 
the radiation frequency (v 0 ) corresponding to 15 [am wavelength. 

This choice is mathematically convenient, is motivated by the fact that a single motion- 
broadened resonance line in a gas atmosphere has a near-Gaussian shape, and gives a fair 
though approximate representation of the actual resonant absorption cross section for 
C02 in the atmosphere. 

The Gaussian cross section is written: 

2 2 

G(v) = c m exp[-(v-v 0 ) / 2ca ] ... (eq.16) 

where a m is the (maximum) absorption cross section at resonance (at v 0 ) and co is the 
Gaussian width of the cross section function. Note: I am using total sample (atmosphere) 
intrinsic cross section, not specific cross section on a per-molecule or per-mass of gas 
basis. The Gaussian function is such that the half width at half maximum (HWHM) of the 
cross section function (intrinsic absorption band) is related to a> as: 



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HWHM = (2 ln(2)Y u co ... (eq.17) 

Next, we find the needed frequency-width of the region of absorption saturation by 
setting G(v) equal to the cross section a e defined to be the cross section at which the C02 
longwave absorption becomes effectively saturated in the atmosphere. That is, we set 
G(v) = c e and we solve for the two absorption band edge positions in frequency v, on 
either side of the central resonance frequency v 0 . 

This gives a saturation band full width as: 

Av = 2(0 [2 /«(o m /c e )] 1/2 ... (eq.18) 

Here, a e is a constant property of a C02-bearing Earth atmosphere and c m , by definition, 
is directly proportional to the atmospheric concentration of C02. Also c m /c e ~ 10 4 for 
C02 at Earth concentrations ([12]: Fig. 2, using intrinsic specific not total cross section). 
The precise value of the latter ratio does not significantly impact our calculations because 
it appears as the argument of the logarithmic function. 

We then examine the variation (8(Av)) of Av (eq. 1 8) with a m and obtain: 

8(Av) / Av = [ 2 ln(oJo e ) J" 1 5(a m )/o m ... (eq.19) 

where 8(a m ) is the considered variation or change in c m . Next, we note that: 
8(Av) / Av = -5(a at ) / (1 - a at ) ... (eq.20) 

since the saturation band width, by definition, negatively and proportionally affects the 
relevant C02 longwave absorption through the atmosphere (nothing within the saturation 
width escapes through any thick layer of atmosphere in our simplified approach), and 

m F co 2 5(C co2 )/C co2 ... (eq.21) 

where F co2 is the present fraction (from 0 to 1) of all greenhouse effects that arise from 
C02. Eq.21 follows from the linear proportionality of cross section with greenhouse 
effect gas concentration for a given gas. 

Therefore we need F co2 . It can most reliably be obtained from satellite spectral 
measurements. This was done in [13] where F co2 ~ 0.26 (for clear sky conditions). 

Given equations such as eq.7 and eq.13 for a where a is defined as a = <s> a T a 4 , it 
follows that changes in Earth surface temperature T a are related to changes in a (arising 
from changes in a a t) as: 

8(T„) = (l/4)T„8(a)/<x...(eq.22) 



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Using the fact that a at is much smaller than one, it follows that for our double-layer 
atmosphere model (eq.13) that changes in surface temperature are related to changes in 
C02 concentration as: 

8(T„) = (1/4) T a [(2 - 2a at )/(3 - 4a at )] [ 2 /n(o m /a e ) f F co2 8(C co2 )/C co2 ... (eq.23) 

where T„ = 290 K, a at = 0.10, a m /a e ~ 10 4 , and F co2 ~ 0.26. In addition, 8(C c0 2)/C co2 = 1 for 
a doubling of the C02 concentration from the present values and 8(Cco2)/C co2 = -0.28 for 
a rolling back of the C02 concentration to the pre-industrial value (from 390 ppmv to 
280 ppmv). 

This immediately gives: 

8(T„) (doubling C02) = +1.4 K ... (eq.24a) 
8(T a ) (pre-industrial C02) = 0.4 K ... (eq.24b) 

Regarding relative temperature change sensitivities of the various parameters, since the 
solar constant itself varies by 6.7% of its mean value over the course of the seasons, let 
6.7% variations be our standard of variation, for the sake of comparisons. 

For our double-layer atmosphere model (eq.13), the temperature change sensitivities for 
6.7% changes in solar constant, etc., are as follows: 

8(T a ) (+6.7% C02) = +0.09K... (eq.25a) 

8(To) (+6.7% solar constant) = +4.9 K ... (eq.25b) 

8(T„) (+6.7% Earth albedo) = -1.5 K ... (eq.25c) 

The latter calculations assume the terms C and D in eq.13 (thermals and water cycle) to 
be proportional to the intensity of solar radiant energy absorbed by Earth. 

The radiation balance steady state temperature of Earth's surface is approximately two 
orders of magnitude more sensitive to changes in solar constant and planetary albedo than 
to changes of atmospheric concentration of greenhouse effect C02. 

Virtually the same results (as eqs.24 and 25) are obtained for our single-layer atmosphere 
model (eq.7) and the same results were previously obtained for a model where the 
atmosphere was treated as an inert (non-thermalizing and non-radiative) infra-red 
greenhouse filter (i.e., like a pane of greenhouse glass that acts only to transmit or reflect 
back some fraction of the longwave emissions from the planet surface) [10]. 



CONCLUDING REMARKS 

Implications for climate science funding and practice 



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In view of the above model sensitivity calculations and given the physical simplicity of 
the model with no free parameters and based on established physical principles it is clear 
that many factors will have a larger effect on surface-temperature-determining radiation 
balance than C02 concentration in the atmosphere. For example, such factors as changes 
in albedo from aerial mineral dust variations due to land use changes, changes affecting 
cloud dynamics (albedo), effective solar irradiance variations, and many more, are 
expected to have larger impacts than C02 concentration under present saturation 
absorption conditions. 

Anyone wishing to focuss on C02 concentration as a climate driver should have the onus 
to explain ignoring the above straightforward demonstration of an approximately two 
order of magnitude irrelevance of C02 relative to solar irradiance (of known seasonal 
variation) and albedo and emissivity (both under-studied and significantly more 
complicated than the effect of C02). 

Regarding the above relative sensitivities, it stands to reason, if reason matters and if we 
are concerned about the global mean surface temperature, that more research funding 
should go into studying solar irradiation variations, regional/planetary shortwave planet 
albedo and regional longwave surface emissivity (dependent on water-contents) rather 
than trying to deduce the relatively subtle effects of changes in "longwave radiative 
forcing" from C02. After all, large scale human land use changes can have dramatic 
effects on both surface radiative properties and colloidal atmospheric particle pollution 
concentrations and depositions, not to mention clouds, clouds, and clouds. 

Likewise, land use practices should be subject to much more scrutiny, if radiation 
warming is our concern, than C02 fluxes into and out of the atmosphere. 

In addition, the tenuous practice of assuming a positive water vapour feedback in models 
would need rigorous validation, despite recent overly optimistic opinionating [14]. See 
section below: "On climate feedbacks and ignorance". 

I have extensively argued from both the social science and climate science perspectives 
that global warming should not be our concern regarding environmental destruction and 
social injustice [4] [15]. 



Relevance to the dominant climate science narrative 

Recently, I critically reviewed the dominant narrative of climate science on several of its 
central points [16]: That the post- industrial atmospheric C02 concentration increase is 
directly a result of fossil fuel burning production of C02, that the post-industrial increase 
in atmospheric C02 concentration causes a greenhouse warming, that a measurable 
global mean surface warming has occurred in the post-industrial period, and that 
anthropogenic global warming radiative forcing drives "climate chaos" and produces 
extreme weather events. 



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The considerations of the present model are mostly consistent with the critical review of 
reference [16]. 

The global inter-carbon-pool flux dynamics of exchanges with the atmosphere and the 
factors affecting these inter-compartment fluxes remain the dominant determinants of the 
resulting atmospheric C02 concentration value [16] (and references therein). 

There remains a vehement debate among atmospheric physicists on the question of 
whether or not a planetary greenhouse effect can occur on a real planet having a 
greenhouse effect gas bearing atmosphere [16] [17] [18] [19] [20]. However, the simplicity 
and robustness of the models developed in the present paper imply that scientists 
claiming an absence of a planetary greenhouse effect mechanism are probably wrong and 
should have the onus to provide a simple physical explanation regarding their proposed 
absence of a planetary greenhouse effect in models of the radiation balance with a 
realistic atmosphere. 

The physical-measurement and mathematical-statistics difficulties in obtaining a mean 
global surface temperature and in estimating the uncertainty error in this mean global 
surface temperature remain [4][16][21]. 

The reality of a post- industrial increase in extreme weather events remains 
undemonstrated and highly contested by climatologists and the physical mechanism 
whereby "climate chaos" would result from extra-C02 "greenhouse radiative forcing" is 
at the level of a tenuous theoretical fantasy [16]. 

Sceptics need to correct many of their statements 

Sceptics are correct that warming alarmism has not been justified from scientific 
principles or from empirical facts. Sceptics are correct that warming alarmism seems to 
be motivated by careerism and corporate/finance opportunism [4] [22]. How else can one 
understand the climate warming feeding frenzy that has too long dominated the climate 
modellers and their entourage of proxy data masseurs, given the straightforward physics 
of the actual temperature sensitivity calculations? 

However, given this same straightforward physics, sceptics (including me) need to stop 
saying things like: 

• "C02 is only a trace gas." Yes, but that is not relevant. What is relevant is C02's 
contribution to the atmosphere's longwave absorption. It is a question of actual 
cross section, not absolute concentration. Satelite spectroscopic measurements are 
unambiguous that C02 contributes 1/4 to 1/3 of all longwave absorption by the 
atmosphere (the rest being due to water vapour and clouds, depending on sky 
conditions) and that C02 absorption is saturated in its main absorption band. 



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• "It's not principally a radiation balance effect." Turn off the Sun and calculate 
Earth's temperature! 

• "Heating the surface by a greenhouse effect violates thermodynamics." No it does 
not. Energy is not lost or created anywhere and local temperatures adjust towards 
steady state to balance energy fluxes. Period. 

• "There is no such thing as a greenhouse effect." While it is true that a 
"greenhouse" with longwave-transparent glass would heat up by retaining the air 
heated by shortwave absorption inside the "greenhouse" it is also true that it heats 
up faster and to greater temperatures if the glass is longwave-opaque. A planet's 
surface (and atmosphere) heats up without any greenhouse gas present but it heats 
up faster and reaches higher temperatures with greenhouse gases. 

And, well, warmists on the other hand need to simply stop and look for intellectual 
honesty. The group think is frightening and needs a serious injection of loss of funding, 
or at least diversification of funding. No one should be allowed to publish anything more 
without making public all the raw data, all the data manipulations, and all the computer 
codes, with manuals and explanations. Science needs to be verifiable. It must never be 
based on authority and protected access. 

On climate feedbacks and ignorance 

It has been shown that one-dimensional dynamical models (vertical dimension only with 
average horizontal layers) of the Earth climate system are as good as three-dimensional 
global circulation models (GCMs) for making sensitivity and climate scenario 
calculations [23]. In many regards they are better because changes in underlying physical 
assumptions are more easily implemented and calculated [23]. Similarly, the present 
calculations show that simple steady state energy balance models also provide the same 
global mean climate predictions and climate sensitivity estimates as one-dimensional 
dynamical models and GCMs; as they should, given the correctly relevant underlying 
physics. 

In all cases, non-negligible surface temperature increases from doubling C02 
concentration are only predicted if some "positive feedback" or amplifying assumption is 
arbitrarily added to the calculations (to "make it work"). 

The warmist scientists have coalesced around the idea and practice of a "positive water 
vapour feedback" in order to resolve the "problem" of small calculated warming impacts 
from increased C02. In my view, the arguments and most heartfelt opinions in support of 
the positive water vapour feedback artifice are tenuous and overly optimistic, typical of 
an edifice seeking to justify itself [14]. 

Regarding such a water vapour feedback, consider the following. More humidity is 
highly correlated to more clouds. Compare winter and summer skies at mid-latitudes, 



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look at the skies over large deserts, etc. Clouds in turn are a major cause of Earth's high 
albedo. As shown above, without clouds and without the atmosphere Earth's mean albedo 
would be -60% less: 0.125 instead of 0.30. This albedo-cloud relation represents a large 
negative feedback potential that has not been sufficiently studied and about which 
geoscientists are mostly ignorant. 

Indeed, geoscientists are mostly ignorant about the mechanisms and geostatistics of 
virtually all the potential feedbacks that could play dominant roles given the dominant 
physical impacts of shortwave Bond albedo and longwave surface and atmosphere 
emissivities. [24] 

What emboldens warmist scientists and modellers, beyond institutional backing and the 
advantages of group think, is the fact that the atmosphere's uniform C02 concentration is 
easy to work with - both in modelling and conceptually - but they should aquire humility 
before indulging their C02 fetish and advancing their tenuous doomsday predictions 
given geoscience's overwhelming ignorance about climate feedbacks. [24] 

And, of course: 

"I argue that by far the most destructive force on the planet is power-driven 
financiers and profit-driven corporations and their cartels backed by military 
might; and that the global warming myth is a red herring that contributes to 
hiding this truth. " [4] [25] 



References and Endnotes 

[1] "Earth's global energy budget" by K.E. Trenberth, J.T. Fasullo, and J. Kiehl. Bulletin 
of the American Meteorological Society, March 2009, 31 1-323, and references therein. 

[2] "Earthshine observations of the Earth's reflectance" by P.R. Goode et al. Geophysical 
Research Letters, 2001, v.28, 1671-1674. 

[3] "Physics: Concepts and connections" by A. Hobson, Prentice Hall, 1999. 

[4] "Global Warming: Truth or dare?" by D.G. Rancourt, Activist Teacher blog, February 
2007. http://activistteacher.blogspot.com/2007/02/global-warming-truth-or-dare.html 

[5] "Greenhouse effect", Wikipedia article, accessed on April 26, 201 1. 
http://en.wikipedia.org/wiki/ Greenhouse_effect 

[6] "Taking measure of the greenhouse effect" AGU Release No. 10-33. 

October 14, 2010. http://www.agu.org/news/press/pr archives/20 1 0/20 1 0-33 .shtml 



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[7] Le Treut, H., R. Somerville, U. Cubasch, Y. Ding, C. Mauritzen, A. Mokssit, T. 
Peterson and M. Prather (plus 26 other contributing authors and two review editors), 
2007: "Historical Overview of Climate Change." In: Climate Change 2007: The Physical 
Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the 
Intergovernmental Panel on Climate Change [Solomon, S., D. Qin, M. Manning, Z. 
Chen, M. Marquis, K.B. Averyt, M. Tignor and H.L. Miller (eds.)]. Cambridge 
University Press, Cambridge, United Kingdom and New York, NY, USA. 

[8] "Water vapour: feedback or forcing?" by GA. Schmidt, RealClimate.org, 2004. (And 
other articles on the site.) 

http://www.realclimate.org/index.php/archives/2004/12/gavin-schmidt/ 

[9] "Evidence of Low Land Surface Thermal Infrared Emissivity in the Presence of Dry 
Vegetation" by Albert Olioso, Guillem Soria, Jose Sobrino, and Benoit Duchemin. IEEE 
Geoscience and Remote Sensing Letters 4, 1 (2007) 1 12-116. 

[10] "Radiation physics constraints on global warming - Revised" by Denis G. Rancourt. 
Posted on the Climate-Guy blog May 12, 201 1. 

http://climateguy.blogspot.com/201 l/05/radiation-physics-constraints-on-global_12.html 
http://ia600607.us.archive.org/23/items/RadiationPhysicsConstraintsOnGlobalWarming/ 
RadiationPhysicsConstraintsOnGlobalWarmingfirst-revised-3.pdf 

[11] "Surface air temperature and its changes over the past 150 years" by P.D. Jones et al. 
Reviews of Geophysics, 1999, v.37, 173-199. 

[12] "Infrared radiation and planetary temperature" by Raymond T. Pierrehumbert. 
Physics Today, January 201 1, 33-38. 

http://geosci.uchicago.edu/~rtpl/papers/PhysTodayRT201 1 .pdf 

[13] "Earth's annual global mean energy budget" by J.T. Kiehl and K.E. Trenberth. 
Bulletin of the American Meteorological Society, February 1997, v. 78, 197-208, and 
references therein. 



[14] "A matter of humidity" by A.E. Dessler and S.C. Sherwood. Science, February 
2009, v.323, 1020-1021. 

http :// geotest.tamu. edu/userfiles/2 1 6/ dessler09 .pdf 

[15] "Questioning Climate Politics" by Dru Oja Jay (an interview with Denis Rancourt). 
The Dominion magazine, April 11, 2007. 
http://www.dominionpaper.ca/articles/l 110 

[16] "On the gargantuan lie of climate change science" by Denis G. Rancourt, March 
201 1, Activist Teacher blog. 

http://activistteacher.blogspot.com/201 1/03/on-gargantuan-lie-of-climate-change.html 



Rancourt on radiation physics - C02 little effect Page 20 of 22 



[17] "Falsification Of the atmospheric C02 greenhouse effects within the frame Of 
Physics" by Gerhard Gerlich and Ralf D. Tscheuschner; International Journal of Modern 
Physics B, Vol. 23, No. 3 (2009) pages 275-364. 
http://arxiv.org/PS cache/arxiv/pdf/0707/0707. 1 161 v4.pdf 

[18] "Proof of the atmospheric greenhouse effect" by Arthur P. Smith; 
arXiv:0802.4324vl [physics.ao-ph] 

http://arxiv.org/PS cache/arxiv/pdf/0802/0802.4324vl .pdf 

[19] "Comments on the "Proof of the atmospheric greenhouse effect" by Arthur P. Smith" 
by Gerhard Kramm, Ralph Dlugi, and Michael Zelger; arXiv:0904.2767v3 [physics. ao- 
ph] 

http://arxiv.org/ftp/arxiv/papers/0904/0904.2767.pdf 

[20] "Reply to 'Comment on 'Falsification Of the atmospheric C02 greenhouse effects 
within the frame Of Physics' by Joshua B. Halpern, Chistopher M. Colose, Chris Ho- 
Stuart, Joel D. Shore, Arthur P. Smith, Jorg Zimmermann" by Gerhard Gerlich and Ralf 
D. Tscheuschner, International Journal of Modern Physics B, Vol. 24, No. 10 (2010) 
pages 1333-1359. 

http://www.skvfall.fr/wp-content/gerlich-replv-to-halpern.pdf 

[21] Christopher Essex, Ross McKitrick, Bjarne Andresen. "Does a Global Temperature 
Exist?" Journal of Non-Equilibrium Thermodynamics, Feb 2007, Vol.32, No.l, pages 1- 
27. 

http://www.reference-global.com/doi/abs/10.1515/JNETDY.2007.001 
https://www.cfa.harvard.edu/~wsoon/ArmstrongGreenSoon08-Anatomy- 
d/EssexMcKitrickAndresen07-globalT JNET2007.pdf 

[22] "The Corporate Climate Coup" by David F. Noble. Posted on the Activist-Teacher 
blog on May 1,2007. 

http://activistteacher.blogspot.com/2007/05/dgr-in-my-article-entitled-global.html 

[23] "An Introduction to Simple Climate Models used in the IPCC Second Assessment 
Report", IPCC Technical Paper II, IPCC, Working Group I, February 1997. Lead 
authors: D. Harvey, J. Gregory, M. Hoffert, A. Jain, M. Lai, R. Leemans, S. Raper, T. 
Wigley, J. de Wolde; Edited by: J.T. Houghton et al. 
http://www.ipcc.ch/pdf/technical-papers/paper-II-en.pdf 

[24] The ignorance in question has been repeatedly pointed out by MIT atmospheric 
physicist Richard S. Lindzen. For example: "Deconstructing global warming", October 
26, 2009, presentation slides. 

http://www.globalwarming.org/wp-content/uploads/2009/10/lindzen-talk-pdfpdf 

[25] "Some big lies of science" by Denis G. Rancourt. Posted at ActivistTeacher on June 
8,2010. 

http://activistteacher.blogspot.com/2010/06/some-big-lies-of-science.html 



Rancourt on radiation physics - C02 little effect Page 21 of 22 



This research was opposed by the University of Ottawa: 

http://uofowatchMogspot.com/2010/09/court-ordered-released-document-shows.html 

Denis G. Rancourt is a former tenured and full professor of physics at the University of 
Ottawa in Canada. He practiced several areas of science (including physics and 
environmental science) which were funded by a national agency and ran an 
internationally recognized laboratory. He has published over 100 articles in leading 
scientific journals and several social commentary essays. He developed popular activism 
courses and was an outspoken critic of the university administration and a defender of 
student and Palestinian rights. He was fired for his dissidence in 2009. His dismissal 
case court hearings are presently on-going and will extend into 2012. 

http://rancourt.academicfreedom.ca/ 



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